Dedekind-Hasse valuation

If D is an integral domainMathworldPlanetmath then it is a PID iff it has a Dedekind-Hasse valuation, that is, a function ν:D-{0}+ such that for any a,bD-{0} either

  • a(b)


  • α(a)β(b)[0<ν(α+β)<ν(b)]

Proof: First, let ν be a Dedekind-Hasse valuation and let I be an ideal of an integral domain D. Take some bI with ν(b) minimal (this exists because the integers are well-ordered) and some aI such that a0. I must contain both (a) and (b), and since it is closed under addition, α+βI for any α(a),β(b).

Since ν(b) is minimal, the second possibility above is ruled out, so it follows that a(b). But this holds for any aI, so I=(b), and therefore every ideal is princple.

For the converse, let D be a PID. Then define ν(u)=1 for any unit. Any non-zero, non-unit can be factored into a finite product of irreduciblesPlanetmathPlanetmath (since PID is a UFD), and every such factorization of a is of the same length, r. So for aD, a non-zero non-unit, let ν(a)=r+1. Obviously r+.

Then take any a,bD-{0} and suppose a(b). Then take the ideal of elements of the form {α+β|α(a),β(b)}. Since this is a PID, it is a principal idealMathworldPlanetmathPlanetmathPlanetmath (c) for some rD-{0}, and since 0+b=b(c), there is some non-unit xD such that xc=b. Then N(b)=N(xr). But since x is not a unit, the factorization of b must be longer than the factorization of c, so ν(b)>ν(c), so ν is a Dedekind-Hasse valuation.

Title Dedekind-Hasse valuation
Canonical name DedekindHasseValuation
Date of creation 2013-03-22 12:51:16
Last modified on 2013-03-22 12:51:16
Owner Henry (455)
Last modified by Henry (455)
Numerical id 5
Author Henry (455)
Entry type Definition
Classification msc 13G05
Related topic EuclideanValuation
Defines Dedekind-Hasse norm
Defines Dedekind-Hasse valuation